Abstract: Monte Carlo Methods (MCMs) have been used extensively in diverse computational applications in the sciences, engineering, and finance. This is due to their natural parallelism, data parsimony and locality, and their capability to tackle high dimension problems that are otherwise intractable. In this mini-symposium, we will present several talks that study the use of MCMs to solve partial differential equations (PDEs). These include using the Feynman-Kac formula to develop MCMs for PDEs, using polynomial chaos for solving stochastic PDEs, Monte Carlo linear solvers that arise from PDEs, algorithmic issues of the walk-on-sphere method, fault tolerance in multilevel MCMs, stability analysis of MCMs for mixed type PDEs, estimation of diffusion process sensitivities, as well as the application of MCMs in capacitance calculation of microchip ICs and multi-asset finance options.

MS-Th-BC-35-210:30--11:00Stochastic collocation method for solving PDEs with random coefficientsShalimova, Irina (ICM&MG SB RAS)Sabelfeld, Karl (Inst. of computational mathematics & mathematical geophysics, Russian Acad. of Sci.)Abstract: We develop a technique based on a polynomial chaos expansion for solving Darcy equation in stochastic porous media. For the input hydraulic conductivity random field we use Karhunen-Loeve expansion. To determine the coefficients of the polynomial chaos expansion we use probabilistic collocation method. We present the numerical results for different Eulerian and Lagrangian statistical characteristics of the flow calculated by both Monte Carlo and probabilistic collocation methods.

MS-Th-BC-35-311:00--11:30Stability issues in MC integration of SDEsPetersen, Wes (ETH Zurich)Abstract: One often encounters difficulties in Monte-Carlo
simulations of complex stochastic processes. For example,
sometimes hyperbolic components appear in a formal diffusion
matrix. In this talk, two issues will be discussed: (1) keeping
oscillating functionals of complex processes on stable orbits,
and (2) filtering away unstable hyperbolic components. In the
first situation, we will examine simple complex linear
oscillators. The second case will be a coherent states
representation for the Bose-Einstein Hamiltonian treated as
quantum noise.